Two-dimensional conformal field theory and beyond. Lessons from a continuing fashion

Abstract

Two-dimensional conformal field theory (CFT) has several sources: the search for simple examples of quantum field theory, the description of surface critical phenomena, the study of (super)string vacua. In the present overview of the subject we emphasize the role of CFT in bridging the gap between mathematics and quantum field theory and discuss some new physical concepts that emerged in the study of CFT models: anomalous dimensions, rational CFT, braid group statistics. In an aside, at the end of the paper, we share the misgivings, recently expressed by Penrose, about some dominant trends in fundamental theoretical physics.